Question: Estimating $e^{1.45}$ using a Taylor polynomial about $x=2$, what is the least degree of the polynomial that assures an error smaller than $0.001$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $2$ (Choice B) B $3$ (Choice C) C $4$ (Choice D) D $5$
We will use the Lagrange error bound. Let's assume the polynomial's degree is $n$. The $(n+1)^{\text{th}}$ derivative of $e^x$ is $e^x$. On the interval between $x=1.45$ and $x=2$, the greatest value of the derivative is $e^{2}\approx7.4$. The Lagrange bound for the error assures that: $|R_n(1.45)|\leq \left| \dfrac{7.4}{(n+1)!}(1.45-2)^{n+1} \right|$ Solving $\dfrac{7.4}{(n+1)!}0.55^{n+1}<0.001$ using trial and error, we find that $n\geq5$. In conclusion, the least degree of the polynomial that assures our error bound is $5$.